One reason why this is an interesting invariant of an elliptic differential operator is that when deforming the operator by a compact operator then the dimension of both the kernel and the cokernel may change, but their difference remains the same. Hence one may think of the analytic index as a “corrected” version of its kernel, such as to make it be more invariant.

Proof

The last step here follows from an argument which is as simple as it is paramount whenever anything involves supersymmetry:

the point is that if a (hermitean) operator HH has a superchargeDD, in that H=D2H = D^2, then all its non-vanishing eigenstates appear in “supermultiplet” pairs of the same eigenvalue: if |ψ⟩|\psi\rangle has eigenvalueE>0E \gt 0 under HH, then

Therefore all eigenstates for non-vanishing eigenvalues appear in pairs whose members have opposite sign under the supertrace. So only states with H|ψ⟩=0H |\psi\rangle = 0 contribute to the supertrace. But if HH and DD are hermitean operators for a non-degenerate inner product, then it follows that (D2|ψ⟩=0)⇔(D|ψ⟩=0)(D^2 |\psi\rangle = 0) \Leftrightarrow (D|\psi\rangle = 0) and hence these are precisely the states which are also annihilated by the supercharge (are in the kernel of DD), hence are precisely only the supersymmetric states.

On these now the weight exp(−tD2)=1\exp(- t D^2) = 1 and hence the supertrace over this “Euclidean propagator” simply counts the number of supersymmetric states, signed by their fermion number.

And hence even more generally one may regard any composition in KKKK as as a generalized index map. Via the universal characterizatin of KKKK itself, this then gives a fundamental and general abstract characterization of the notion of index: